Proposition C1 (Reverse Chernoff-Cramer Bound [1]}) 

Let \(Z\)  be a random variable with moment generating function \(\phi _Z(\lambda ) = \mathbb {E} \exp (\lambda Z)\)  defined on a set \(\Lambda \in \mathbb {R}\) . For any \(t>0\) , we have the following lower bounds:
\(&\mathbb {P}(Z \ge t) \ge \nonumber \\&\sup _{\alpha , \beta > 1, \lambda > \lambda ^{\prime } \ge 0, \atop \lambda \alpha \in \Lambda }\lbrace  \phi _Z(\lambda ) \exp (-\beta \lambda t) - \phi _Z(\alpha \lambda ) \exp (-\alpha \beta \lambda t) - \exp (-(\beta \lambda - \lambda ^{\prime })t)\phi _Z(\lambda - \lambda ^{\prime }) \rbrace , \\&\mathbb {P}(Z \le -t) \ge \nonumber \\& \sup _{\alpha , \beta > 1, \lambda > \lambda ^{\prime } \ge 0, \atop -\lambda \alpha \in \Lambda }\lbrace  \phi _Z(-\lambda ) \exp (-\beta \lambda t) - \phi _Z(-\alpha \lambda ) \exp (-\alpha \beta \lambda t) - \exp (-(\beta \lambda - \lambda ^{\prime })t)\phi _Z(\lambda ^{\prime } - \lambda ) \rbrace .\) 

